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Boundary singularities of positive solutions of quasilinear Hamilton-Jacobi equations (1412.4613v5)
Published 15 Dec 2014 in math.AP
Abstract: We study the boundary behaviour of the solutions of (E) $-\Delta_p u+|\nabla u|q=0$ in a domain $\Omega \subset \mathbb{R}N$, when $N\geq p > q >p-1$. We show the existence of a critical exponent $q_* < p$ such that if $p-1 < q < q_$ there exist positive solutions of (E) with an isolated singularity on $\partial\Omega$ and that these solutions belong to two different classes of singular solutions. If $q_\leq q < p$ no such solution exists and actually any boundary isolated singularity of a positive solution of (E) is removable. We prove that all the singular positive solutions are classified according the two types of singular solutions that we have constructed.